Hyperchaotic Circuit Based on Memristor Feedback with Multistability and Symmetries

A novel hyperchaotic circuit is proposed by introducing a memristor feedback in a simple Lorenz‐like chaotic system. Dynamic analysis shows that it has infinite equilibrium points and multistability. Additionally, the symmetrical coexistent attractors are investigated. Further, the hyperchaotic system is implemented by analogue circuits. Corresponding experimental results are completely consistent with the theoretical analysis.


Introduction
Chua [1] predicted the existence of the fourth basic circuit element memristor in 1971. However, it was not until 2008 that a team of researchers at Hewlett-Packard (HP) made nano-membrane memristive devices for the first time [2]. Since then, memristors have attracted great attentions and relative researches cover academia and industry. e emergence of memristive devices is expected to achieve nonvolatile memory. Moreover, memristors are considered to be a better way to implement artificial neural network synapses in hardware [3]. Also, due to the nonlinear characteristic of memristor, it can be used in chaotic circuits.
In 2008, Chua and Itoh [4] first constructed memristor oscillators from Chua's oscillators by replacing Chua's diodes with memristors, where the piece-wise linear (PWL) memristor model was used. Two years later, Muthuswamy [5] applied the smooth cubic flux-controlled memristor model to Chua's oscillators, which generated different chaotic attractors. e same memristor model was used in [6,7] too. More specifically, in 2011, a twin-T notch filter was used to control chaos in a memristor based circuit in [6], and [7] implemented a hyperchaotic circuit by using this model in 2012. In 2019, based on a new current-controlled memristor, a new fourdimensional chaotic circuit is proposed and studied [8]. In these studies, nonlinear characteristic of memristor and dynamical properties of memristive chaotic and hyperchaotic circuits had been researched. In the same year, Wang presented a novel memristor model based on light dependent resistor (LDR) and its emulator in [9] and further realized a memcapacitor emulator based on the LDR memristor in [10]. Later in 2014, [11] analyzed dynamic characteristics of a LDR memristor based chaotic system. Another new memristor model was applied to Chua's circuit, and the complex transient dynamic behaviour was found in this circuit [12]. Up to now, only the HP memristor can be manufactured and widely used. In order to research application circuits with the HP memristor, [13] presented a flux-controlled model of HP memristor and analyzed its chaotic oscillator. A chaotic circuit based on two HP memristor in antiparallel was presented in [14]. e above chaotic circuits are simple and use less numbers of circuit elements. Nevertheless, there are other chaotic circuit design approaches, like Lorenz system [15], Chen system [16], Lü system [17] and most other nonlinear dynamical systems [18] could be described by ordinary differential equations containing product terms, which could easily be realized by electronic circuits via standard operational amplifiers (op-amps) and four-quadrant analogue multipliers. Generally, the op-amp is used for the integral operation and nonlinear multiplicative operation is implemented by multipliers. In this paper, the op-amps and multipliers are working together with a memristor to introduce a feedback in a chaotic circuit to realize a hyperchaotic circuit.
In this paper, based on a simple chaotic circuit of three dimension Lorenz-like system, a hyperchaotic circuit was constructed by introducing a memristor feedback. is paper is organized as follows. Section 2 gives the memristor mathematical model and its equivalent circuit. In Section 3, the hyperchaotic circuit is constructed, the Lyapunov exponent and the Lyapunov dimension of the system are calculated. Section 4 discusses the stability and equilibrium points of the system. Section 5 analyzes the dynamic properties of the system, including the power spectrum and symmetrical coexistent attractors of the system. Section 6 presents the results of the circuit experiment. Section 7 draws conclusions.

Memristor Model and Its Equivalent Circuits
According to [19], memristors can be classified as ideal memristors, ideal generic memristors, generic memristors, and extended memristors, and the current controlled and voltage controlled ideal memristor can be described by the following relations: where and denote the voltage cross and current goes through a memristor, and are the charge and flux on the memristor at time , and represent memristance (1) 푣 = 푅 푞 푖, 푑푞/푑푡 = 푖, (2) 푖 = 퐺 휑 푣, 푑휑/푑푡 = 푣, and memductance. A smooth flux-controlled memristor model was researched in [5], whose memductance is given by: where and are constants and setting as 훼 = 1, 훽 = 0.02 through the whole paper. Absolutely, Equation (3) describes an ideal memristor. By Matlab simulation, Figure 1(a) shows the voltage-current characteristics of the memristor at different frequencies, when it is powered by a sinusoidal voltage source, whose amplitude is chosen as 4 V and the angular frequency is setting to 0.1 rad/s. From Figure 1(a), we can see that the memristor degenerates into a normal resistor when the angular frequency is large enough. e time domain diagrams of voltage and current are shown in Figure 1    3 Complexity e emulator of the memristor described by Equation (3) is given in Figure 3. A er the input voltage passing through an integrator consisting of a resistor R 1 , a capacitor C 1 and an op-amp U 1 , the negative flux(−휑) is gotten at node 1. en 2 could be obtained by multiplier A 1 and the response of an inverting adder consists of resistors R 2 , R 3 , R 4 and op-amp U 2 can be described by the following equation: where 1 is a DC voltage bias. e signal −퐺 휑 at node 2 is inverted by the inverter composed by R 5 , R 6 and op-amp U 3 , then a er at node 3 multiplying with the input voltage v in multiplier A 2 , the corresponding output current i could be obtained. e Multisim simulation results of this equivalent circuit are shown in Figure 4, with input voltage 휈 = 4 sin(2휋푓푡). Figure 4(a) shows the pinched hysteresis loop, the timing diagrams of voltage and current are shown in Figure 4(b), the results verify the effectiveness of the equivalent circuit.

Construction of Hyperchaotic Circuit
A simple Lorenz-like system was reported in [20], which is described as: Such dynamical systems could easily be implemented by the op-amps and multipliers. As shown in Figure 5, if we consider the memristor as an open circuit, i.e., when the feedback has not been applied, it realizes the Lorenz-like system described by Equation (5). Now let the memristor mentioned above be used as a feedback element, according to Kirchhoff 's law, basic circuit theory and the relationship between the terminal voltage and current described by Equations (2) and (3), the equivalent hyperchaotic circuit state equation in Figure 5 can be written as: en the dimensionless state equation of the circuit can be described by: where , , and are internal state variables of the memristive system, and , , , , and are parameters. When we set the parameters as 푎 = 24, 푏 = 4, 푐 = 19, 푑 = 9, 훼 = 1 and 훽 = 0.02, the system can generate hyperchaos. e hyperchaotic attractor as shown in Figure 6, which indicates the hyperchaotic system has a complex dynamic behaviour. Figures 6(a)-6(d) are the hyperchaotic phase diagrams on x-y, x-z, y-z, and x-w planes, respectively. e Lyapunov exponent can be calculated as 퐿퐸 1 = 0.3399, 퐿퐸 2 = 0.2144, 퐿퐸 3 = −0.0045, 퐿퐸 4 = −33.5498 by Jacobi method, and the Lyapunov dimension is 3.0164 calculated by Equation (8), which indicates that the system is hyperchaotic under the appropriate parameters. e expression of divergence of the memristive hyperchaotic system can be described as follows: when parameters 푎 = 24, 푏 = 4, 푐 = 19, 푑 = 9, the ∇푉 is less than 0, so the system is dissipative and converge exponentially.
Hyperchaotic circuit by introducing memristor feedback.  According to the Routh-Hurwitz stability criterion, the system is stable if and only if the sequences of determinants of its principal submatrices are all positive. In other words, if 1 , 2 , 3 and 4 are all greater than 0, then the system is stable. When parameters , , , and are set to 24, 9, 19 and 4, 훥 2 = − 912푒 2 /25 − 5640, which is always less than 0. So all the equilibria of the system are unstable under the above parameters, and the system may generate chaos or hyperchaos.

Poincaré Mapping and Sequence Diagram Analyses.
When parameters are setting as 푎 = 24, 푏 = 4, 푐 = 19, 푑 = 9, the Poincaré mapping of the hyperchaotic system is shown in Figure 7. Figure 7(a) is the Poincare mapping obtained when the cross plane is selected as 푥 = 0. To describe its acquisition process more vividly, Figure 7(b) shows the 3 dimensional stereoscopic graphics. Poincaré mapping reflects the motion characteristics of the system. As shown in Figure 7(a), the Poincaré mapping manifests as dense points and its structure is hierarchical, so it can be determined that the motion of the system is in a chaotic or hyperchaotic state. Figure 8 presents the sequence diagrams of the system, which indicates the motion of the system is pseudo-random and aperiodic. So combining with the above analyses and

Equilibrium Points and Stability of the System
In order to resolve the equilibrium of the system, let the right side of Equation (7) equal to 0, and it is found that the equilibrium points set is 푂 = 푥 = 푦 = 푧 = 0, 푤 = 푒 , which indicates that there are infinite equilibria existing in the memristive system. Different from most of the common chaotic systems, memristive chaotic systems often contain infinite equilibria points. The system's Jacobian matrix at O is described as: Further, the characteristic equation corresponding to the Jacobian matrix can be obtained as follow: From Equation (11), the coefficients of the polynomial can be obtained, which are 푎 0 = 1, 푎 1 = 푎 + 푑, 푎 2 = 푎푑 − 푎푐 − 푏푐 − 0.02푏푐푒 2 , 푎 3 = − 0.02푏푐푑푒 2 + 푎푐푑 + 푏푐푑 , 푎 4 = 0, 푎 5 = 0, 푎 6 = 0 and 푎 7 = 0. As shown in Equation (12), the coefficients of the polynomial could be arranged into a square matrix form, called the Hurwitz matrix.     the Lyapunov exponents, it can be judged that the system could be in the hyperchaotic state under the above parameters.

Influence of Parameter Variation on Dynamics of
System. Parameter variation affects the equilibrium point and stability of the system, so it is meaningful to discuss the impact of parameter changes of the system. Figure 9 shows the corresponding Lyapunov exponent spectrum and bifurcation diagram when parameter is changing from 6 to 18, while other parameters , , are fixed.
It is worth pointing out that there should be four Lyapunov exponents in a four dimension system. However, only three Lyapunov exponents are shown in Figure 9(a), because the fourth Lyapunov exponent is negative and much smaller than the other three ones, so in order to clearly show the details of the Lyapunov exponent spectrum, the fourth Lyapunov exponent is not shown. And Figure 9(b) is the corresponding bifurcation diagram, when the parameter increases, the system gradually transitions from than 0, a Lyapunov exponent equal to 0, and a Lyapunov exponent less than 0 until reaches 11.8. is also proves that the system can indeed be in hyperchaotic state under certain parameters.
hyperchaotic state to periodic state and the bifurcation diagram is consistent with the Lyapunov exponent spectrum. Moreover, it can be clearly observed from Figure 9(a) that during grows, there are two Lyapunov exponents greater  Complexity 8 Figure 11 shows the typical coexisting attractors of this system, the blue and red trajectories are phase diagrams starting from symmetrical initial values under same sets of parameters. Table 1 presents different states and their corresponding parameters, initial values, and figure numbers in Figure 11. More specifically, in Figures 11(a) and 11(b), both of them are chaotic attractor coexisting with other symmetrical chaotic attractors. Figures 11(c) and 11(d) describe two symmetrical hyperchaotic attractors coexisting with each other. Figure 11(e) shows symmetrical quasiperiodic coexistent attractors, and symmetrical period 1 coexistent attractors are shown in Figure 11(f).

Circuit Experiment
From the previous analysis, the hyperchaotic system described by Equation (7)

Power Spectrum Analysis.
As everyone knows, the power spectrum of a periodic signal is a discrete spectrum, the power spectrum of aperiodic signals is a continuous spectrum and for a noise signal, whose power spectrum is continuous and smooth. Chaotic and hyperchaotic signals are a kind of aperiodic signal, so its power spectrum should be a continuous spectrum. Figure  10 shows the attractors and the normalized power spectrums corresponding to different states of the system. Concretely, Figure 10(a) shows period-1 attractor when parameter 푑 = 14.64 and Figure 10(b) presents its sole power spectrum. Similarly, Figure 10(c) is period-2 attractor as 푑 = 12 and Figure 10(d) is the corresponding discrete power spectrum. Figure 10(e) shows the hyperchaotic attractor when 푑 = 9, it is worth to note that there are a large number of peaks in its corresponding power spectrum Figure 10(f). Such phenomenon is mainly due to the existence of numerous double periodic bifurcations in chaos or hyperchaos. It can be found that power spectrum analysis can be effectively used to compare and distinguish noise signals, periodic signals, chaotic and hyperchaotic signals.

Symmetrical Coexistent Attractors.
e system shown in Equation (7) is symmetric about the -axis, which can be derived from invariance when transforming 푥, 푦, 푧, 푤 to −푥, −푦, 푧, −푤 , as shown in Equation (13). erefore, the system can enter symmetrical trajectories under symmetrical initial values. In order to avoid the nonlinear distortion and observe the better waveforms on the oscilloscope, the proportional compression transformation and the time scale transformation are needed to carry out on Equation (7). Let kx → x, ky → y, kz → z and τ → t 0 t, where 푘 = 10 and 푡 0 = 1000, the system can be described as: e designed experimental circuit is shown in Figure 12 and this circuit can be described by: Comparing Equation (15) with Equation (14), the corresponding circuit parameters can be determined as follows: C 1 = C 2 = C 3 = C 4 = 10 nF, R 1 = 100 kΩ, R 2 = R 11 = R 14 = 1 kΩ, R 3 = R 4 = 20 kΩ, R 9 = R 8 = R 12 = R 13 =10 kΩ, R 5 = 2.5 kΩ, R 6 = R 7 = 4.1 kΩ, R 10 = 5 kΩ, R 15 = 11 kΩ. e op-amp LF347 and analogue multipliers AD633 with ±15 V power supplies are used to construct the experimental circuit. e experimental results are shown in Figure 13, which matches well with Matlab numerical simulations shown in Figure 6.

Conclusions
In this paper, a hyperchaotic circuit based on memristor feedback is proposed. In addition to some conventional analysis, the frequency domain characteristics of hyperchaotic signals are analyzed by the power spectrum. Further, the infinite equilibrium points and the symmetrical coexistent attractors are found. Finally, the theoretical analysis was verified by experimental circuits.
Data Availability e data used to support the findings of this study are available from the corresponding author upon request.